Final answer:
Using the substitution x = 7/2tan(t), we can transform the integral I = ∫dx/√(49x² + 4) into a form involving trigonometric identities, which assists in simplifying and calculating the integral.
Step-by-step explanation:
The substitution x = \(\frac{7}{2}\)tan(t) is used to simplify the integral I = \int\frac{dx}{\sqrt{49x^2 + 4}}. To perform this substitution, we will need to also find the differential dx in terms of dt. Using the trigonometric identity for tan(t), this substitution will lead to a more manageable form of the integral. The steps are as follows:
- Determine dx by differentiating x with respect to t.
- Substitute x and dx into the original integral.
- Use the Pythagorean identity to simplify the resulting trigonometric expression.
- Integrate with respect to t.
By doing so, we leverage trigonometric identities and substitutions to evaluate integrals that are otherwise difficult to compute directly.